CONTENTS &
ABSTRACTS

In
English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

** **

Volume 50 No. 3
September 2001

Semiparallel
submanifolds with plane generators of codimension two in a Euclidean space;
115–123

Ülo LUMISTE

**Abstract.** A submanifold generated by plane leaves of codimension two in a
Euclidean space is, in general, intrinsically a Riemannian manifold of
conullity two. All such manifolds have been classified into four classes:
planar, hyperbolic, parabolic, and elliptic, i.e. having, respectively,
infinitely many, two, one, or no real intrinsically asymptotic distributions.
It is proved that if such a submanifold is semiparallel and intrinsically a
manifold of conullity two, then it must be planar. This verifies, for the case
considered here, a conjecture that a semiparallel submanifold, which is
intrinsically of conullity two, must be planar. Validity of this conjecture has
been established previously by the author for the three-dimensional semiparallel
submanifolds.

**Key words:** Riemannian manifolds of conullity two, asymptotic foliations,
semiparallel submanifolds.

Pseudosymmetric
contact metric manifolds in the sense of M. C. Chaki; 124–132

Kadri
ARSLAN, Cengizhan MURATHAN, Cihan ÖZGÜR, and

Ahmet
YILDIZ

**Abstract.** We consider
pseudosymmetric and pseudo Ricci symmetric manifolds in the sense of M. C.
Chaki. The case _{} is assumed to
be a contact metric manifold with _{} belonging to _{}-nullity distribution.

**Key words:** contact
manifolds, Einstein _{}-Einstein) manifolds, _{}-nullity distribution, pseudosymmetric manifolds of
Chaki type.

Fully
discrete collocation method for weakly singular integral equations; 133–144

Enn TAMME

**Abstract.** To find the approximate solutions of a weakly singular integral
equation by the collocation method it is necessary to solve linear systems
whose coefficients are expressed as integrals. These integrals cannot usually
be computed exactly. We get the fully discrete collocation method when we
approximate the integrals by quadrature formulas on nonuniform grid. In this
paper an appropriate grid is formed and the dependence of the convergence rate
of this method on the choice of the grid is studied.

**Key words:** weakly singular integral equation, collocation method.

Fast
solvers of generalized airfoil equation of index –1; 145–154

Gennadi VAINIKKO

**Abstract.** We consider the generalized airfoil equation in the situation
where the index of the problem is –1. We periodize the problem, then discretize
it by a fully discrete version of the trigonometric collocation method and
apply the conjugate gradient method to solve the discretized problem. The
approximate solution appears to be of optimal accuracy in a scale of Sobolev
norms, and the *N* parameters of the
approximate solution can be determined by *O*(*N *log *N*) arithmetical operations.

**Key words:** airfoil equations, fast solvers.

Application
of wavelet transform techniques to vibration studies; 155–168

Ülo LEPIK

**Abstract.** Wavelet transform techniques are applied to analysis of linear
vibrations. It is shown that in some simple cases wavelet transform can be
accomplished analytically. Damped and forced vibrations of single and two
degrees of freedom are considered. The achieved results can be used for
interpreting more complicated cases.

**Key words: **linear vibrations, damped and forced vibrations,
Fourier transform, wavelets.

Properties
of high-temperature superconductors in a two-band model with the doping-dependent
electron spectrum; 169–178

Nikolai KRISTOFFEL and Pavel RUBIN

**Abstract.** A simple model of interband superconductivity, incorporating a
basic band and a doping created one (constant total number of states), is
considered, keeping in mind the two-component scenario of high-*T*_{c} superconductivity. The
dependences of *T*_{c}, its
isotope exponent, superconductivity gaps, etc. on doping are calculated using
an illustrative set of parameters. The observed types of behaviour are
reproduced. Diminishing gap ratios with hole doping have been found. In the
one-particle excitation spectrum the peak–dip–hump type complex can be
identified. In spite of crude approximations the model reflects, in general,
the experimental findings on high-*T*_{c}
cuprates. Two-band models will remain a channel for looking for the high-*T*_{c} mechanism.

**Key words:** high-temperature superconductivity, two-band models, doping,
gaps.

CHRONICLE

Ülo Lepik 80; 179